October 2018, 15(5): 1255-1270. doi: 10.3934/mbe.2018058

The mean and noise of stochastic gene transcription with cell division

1. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

2. 

College of Science, Guangxi University of Science and Technology, Liuzhou 545006, China

3. 

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510275, China

* Corresponding author: Jianshe Yu

Received  January 31, 2018 Revised  April 16, 2018 Published  May 2018

Fund Project: The authors are supported by National Natural Science Foundation of China (11631005, 11461002), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16) and the Innovation Research Grant for the Postgraduates of Guangzhou University (2017GDJC-D01)

Life growth and development are driven by continuous cell divisions. Cell division is a stochastic and complex process. In this paper, we study the impact of cell division on the mean and noise of mRNA numbers by using a two-state stochastic model of transcription. Our results show that the steady-state mRNA noise with symmetric cell division is less than that with binomial inheritance with probability 0.5, but the steady-state mean transcript level with symmetric division is always equal to that with binomial inheritance with probability 0.5. Cell division except random additive inheritance always decreases mean transcript level and increases transcription noise. Inversely, random additive inheritance always increases mean transcript level and decreases transcription noise. We also show that the steady-state mean transcript level (the steady-state mRNA noise) with symmetric cell division or binomial inheritance increases (decreases) with the average cell cycle duration. But the steady-state mean transcript level (the steady-state mRNA noise) with random additive inheritance decreases (increases) with the average cell cycle duration. Our results are confirmed by Gillespie stochastic simulation using plausible parameters.

Citation: Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058
References:
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D. Antunes and A. Singh, Quantifying gene expression variability arising from randomness in cell division times, J. Math. Biol., 71 (2015), 437-463. doi: 10.1007/s00285-014-0811-x.

[2]

A. ArkinJ. Ross and H. H. Mcadams, Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells, Genetics, 149 (1998), 1633-1648.

[3]

P. Bastiaens, Systems biology: When it is time to die, Nature, 459 (2009), 334-335. doi: 10.1038/459334a.

[4]

C. BertoliJ. M. Skotheim and R. A. de Bruin, Control of cell cycle transcription during G1 and S phases, Nat. Rev. Mol. Cell Bio., 14 (2013), 518-528. doi: 10.1038/nrm3629.

[5]

W. J. BlakeM. KærnC. R. Cantor and J. J. Collins, Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637. doi: 10.1038/nature01546.

[6]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Exact and approximate distributions of protein and mRNA in the low-copy regime of gene expression, J. Math. Biol., 64 (2012), 829-854. doi: 10.1007/s00285-011-0433-5.

[7]

A. BrockH. Chang and S. Huang, Non-genetic heterogeneity-a mutation-independent driving force for the somatic evolution of tumours, Nat. Rev. Genet., 10 (2009), 336-342. doi: 10.1038/nrg2556.

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H. H. Chang, et al., Transcriptome-wide noise controls lineage choice in mammalian progenitor cells, Nature, 453 (2008), 544–547. doi: 10.1038/nature06965.

[9]

E. Clayton, et al., A single type of progenitor cell maintains normal epidermis, Nature, 446 (2007), 185–189. doi: 10.1038/nature05574.

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A. Colman-Lerner, et al., Regulated cell-to-cell variation in a cell-fate decision system, Nature, 437 (2005), 699–706.

[11]

M. R. Dowling, et al., Stretched cell cycle model for proliferating lymphocytes, Proc. Natl. Acad. Sci. USA, 111 (2014), 6377–6382. doi: 10.1073/pnas.1322420111.

[12]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919.

[13]

P. L. FelmerA. QuaasM. X. Tang and J. S. Yu, Random dynamics of gene transcription activation in single cells, J. Differ. Equations, 247 (2009), 1796-1816. doi: 10.1016/j.jde.2009.06.006.

[14]

D. Fraser and M. Kærn, A chance at survival: Gene expression noise and phenotypic diversification strategies, Mol. Microbiol., 71 (2009), 1333-1340. doi: 10.1111/j.1365-2958.2009.06605.x.

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I. GoldingJ. PaulssonS. M. Zawilski and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036. doi: 10.1016/j.cell.2005.09.031.

[16]

D. Gonze, Modeling the effect of cell division on genetic oscillators, J. Theor. Biol., 325 (2013), 22-33. doi: 10.1016/j.jtbi.2013.02.001.

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E. D. HawkinsJ. F. MarkhamL. P. Mcguinness and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci. USA, 106 (2009), 13457-13462. doi: 10.1073/pnas.0905629106.

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J. JaruszewiczM. Kimmel and T. Lipniacki, Stability of bacterial toggle switches is enhanced by cell-cycle lengthening by several orders of magnitude, Phys. Rev. E., 89 (2014), 022710. doi: 10.1103/PhysRevE.89.022710.

[23]

F. JiaoM. X. Tang and J. S. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Differ. Equations, 254 (2013), 3307-3328. doi: 10.1016/j.jde.2013.01.019.

[24]

F. JiaoM. X. TangJ. S. Yu and B. Zheng, Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math., 75 (2015), 2396-2420. doi: 10.1137/151005567.

[25]

I. G. Johnston and N. S. Jones, Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions, Proc. R. Soc. A, 471 (2015), 20150050, 19pp. doi: 10.1098/rspa.2015.0050.

[26]

M. KærnT. C. ElstonW. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet., 6 (2005), 451-464.

[27]

S. J. KronC. A. Styles and G. R. Fink, Symmetric cell division in pseudohyphae of the yeast Saccharomyces cerevisiae, Mol. Biol. Cell, 5 (1994), 933-1063. doi: 10.1091/mbc.5.9.1003.

[28]

J. H. KuangM. X. Tang and J. S. Yu, The mean and noise of protein numbers in stochastic gene expression, J. Math. Biol., 67 (2013), 261-291. doi: 10.1007/s00285-012-0551-8.

[29]

E. KussellR. KishonyN. Q. Balaban and S. Leibler, Bacterial persistence: A model of survival in changing environments, Genetics, 169 (2005), 1807-1814. doi: 10.1534/genetics.104.035352.

[30]

K. Lewis, Persister cells, Annu. Rev. Microbiol., 64 (2010), 357-372. doi: 10.1146/annurev.micro.112408.134306.

[31]

Q. Y. LiL. F. Huang and J. S. Yu, Modulation of first-passage time for bursty gene expression via random signals, Math. Biosci. Eng., 14 (2017), 1261-1277. doi: 10.3934/mbe.2017065.

[32]

Y. Y. LiM. X. Tang and J. S. Yu, Transcription dynamics of inducible genes modulated by negative regulations, Math. Med. Biol., 32 (2015), 115-136. doi: 10.1093/imammb/dqt019.

[33]

E. LibbyT. J. Perkins and P. S. Swain, Noisy information processing through transcriptional regulation, Proc. Natl. Acad. Sci. USA, 104 (2007), 7151-7156. doi: 10.1073/pnas.0608963104.

[34]

J. Lloyd-PriceH. Tran and A. S. Ribeiro, Dynamics of small genetic circuits subject to stochastic partitioning in cell division, J. Theor. Biol., 356 (2014), 11-19. doi: 10.1016/j.jtbi.2014.04.018.

[35]

R. Losick and C. Desplan, Stochasticity and cell fate, Science, 320 (2008), 65-68. doi: 10.1126/science.1147888.

[36]

A. A. Martinez and J. M. Brickman, Gene expression heterogeneities in embryonic stem cell populations: Origin and function, Curr. Opin. Cell Biol., 23 (2011), 650-656. doi: 10.1016/j.ceb.2011.09.007.

[37]

B. MunskyG. Neuert and O. A. Van, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187. doi: 10.1126/science.1216379.

[38]

M. OsellaE. Nugent and L. M. Cosentino, Concerted control of Escherichia coli cell division, Proc. Natl. Acad. Sci. USA, 111 (2014), 3431-3435.

[39]

J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. doi: 10.1006/tpbi.1995.1027.

[40]

A Raj and O. A. Van, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell, 135 (2008), 216-226. doi: 10.1016/j.cell.2008.09.050.

[41]

A. SanchezS. Choubey and J. Kondev, Regulation of noise in gene expression, Annu. Rev. Biophys., 42 (2013), 469-491. doi: 10.1146/annurev-biophys-083012-130401.

[42]

A. Singh and L. S. Weinberger, Stochastic gene expression as a molecular switch for viral latency, Curr. Opin. Microbiol., 12 (2009), 460-466. doi: 10.1016/j.mib.2009.06.016.

[43]

S. O. Skinner, et al., Single-cell analysis of transcription kinetics across the cell cycle, eLife, 5 (2016), e12175. doi: 10.7554/eLife.12175.

[44]

L. H. So, et al., General properties of transcriptional time series in Escherichia coli, Nat. Genet., 43 (2011), 554–560. doi: 10.1038/ng.821.

[45]

S. L. Spencer, et al., Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis, Nature, 459 (2009), 428–432. doi: 10.1038/nature08012.

[46]

Q. W. SunM. X. Tang and J. S. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494. doi: 10.1007/s00285-011-0420-x.

[47]

Q. W. SunM. X. Tang and J. S. Yu, Temporal profile of gene transcription noise modulated by cross-talking signal transduction pathways, B. Math. Biol., 74 (2012), 375-398. doi: 10.1007/s11538-011-9683-z.

[48]

P. S. SwainM. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci. USA, 99 (2002), 12795-12800.

[49]

M. X. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol., 253 (2008), 271-280. doi: 10.1016/j.jtbi.2008.03.023.

[50]

M. L. TurnerE. D. Hawkins and P. D. Hodgkin, Quantitative regulation of B cell division destiny by signal strength, J. Immunol., 181 (2008), 374-382. doi: 10.4049/jimmunol.181.1.374.

[51]

Y. VoichekR. Bar-Ziv and N. Barkai, Expression homeostasis during DNA replication, Science, 351 (2016), 1087-1090. doi: 10.1126/science.aad1162.

[52]

H. H. WangZ. J. YuanP. J. Liu and T. S. Zhou, Division time-based amplifiers for stochastic gene expression, Mol. Biosyst., 11 (2015), 2417-2428. doi: 10.1039/C5MB00391A.

[53]

L. S. Weinberger, et al., Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity, Cell, 122 (2005), 169–182. doi: 10.1016/j.cell.2005.06.006.

[54]

J. S. Yu and X. J. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.

[55]

J. S. YuQ. W. Sun and M. X. Tang, The nonlinear dynamics and fluctuations of mRNA levels in cross-talking pathway activated transcription, J. Theor. Biol., 363 (2014), 223-234. doi: 10.1016/j.jtbi.2014.08.024.

show all references

References:
[1]

D. Antunes and A. Singh, Quantifying gene expression variability arising from randomness in cell division times, J. Math. Biol., 71 (2015), 437-463. doi: 10.1007/s00285-014-0811-x.

[2]

A. ArkinJ. Ross and H. H. Mcadams, Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells, Genetics, 149 (1998), 1633-1648.

[3]

P. Bastiaens, Systems biology: When it is time to die, Nature, 459 (2009), 334-335. doi: 10.1038/459334a.

[4]

C. BertoliJ. M. Skotheim and R. A. de Bruin, Control of cell cycle transcription during G1 and S phases, Nat. Rev. Mol. Cell Bio., 14 (2013), 518-528. doi: 10.1038/nrm3629.

[5]

W. J. BlakeM. KærnC. R. Cantor and J. J. Collins, Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637. doi: 10.1038/nature01546.

[6]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Exact and approximate distributions of protein and mRNA in the low-copy regime of gene expression, J. Math. Biol., 64 (2012), 829-854. doi: 10.1007/s00285-011-0433-5.

[7]

A. BrockH. Chang and S. Huang, Non-genetic heterogeneity-a mutation-independent driving force for the somatic evolution of tumours, Nat. Rev. Genet., 10 (2009), 336-342. doi: 10.1038/nrg2556.

[8]

H. H. Chang, et al., Transcriptome-wide noise controls lineage choice in mammalian progenitor cells, Nature, 453 (2008), 544–547. doi: 10.1038/nature06965.

[9]

E. Clayton, et al., A single type of progenitor cell maintains normal epidermis, Nature, 446 (2007), 185–189. doi: 10.1038/nature05574.

[10]

A. Colman-Lerner, et al., Regulated cell-to-cell variation in a cell-fate decision system, Nature, 437 (2005), 699–706.

[11]

M. R. Dowling, et al., Stretched cell cycle model for proliferating lymphocytes, Proc. Natl. Acad. Sci. USA, 111 (2014), 6377–6382. doi: 10.1073/pnas.1322420111.

[12]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919.

[13]

P. L. FelmerA. QuaasM. X. Tang and J. S. Yu, Random dynamics of gene transcription activation in single cells, J. Differ. Equations, 247 (2009), 1796-1816. doi: 10.1016/j.jde.2009.06.006.

[14]

D. Fraser and M. Kærn, A chance at survival: Gene expression noise and phenotypic diversification strategies, Mol. Microbiol., 71 (2009), 1333-1340. doi: 10.1111/j.1365-2958.2009.06605.x.

[15]

I. GoldingJ. PaulssonS. M. Zawilski and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036. doi: 10.1016/j.cell.2005.09.031.

[16]

D. Gonze, Modeling the effect of cell division on genetic oscillators, J. Theor. Biol., 325 (2013), 22-33. doi: 10.1016/j.jtbi.2013.02.001.

[17]

E. D. HawkinsJ. F. MarkhamL. P. Mcguinness and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci. USA, 106 (2009), 13457-13462. doi: 10.1073/pnas.0905629106.

[18]

E. D. Hawkins, et al., A model of immune regulation as a consequence of randomized lymphocyte division and death times, Proc. Natl. Acad. Sci. USA, 104 (2007), 5032–5037. doi: 10.1073/pnas.0700026104.

[19]

L. F. Huang, et al., The free-energy cost of interaction between DNA loops, Sci Rep-UK, 7 (2017). doi: 10.1038/s41598-017-12765-x.

[20]

D. Huh and J. Paulsson, Non-genetic heterogeneity from random partitioning at cell division, Nat. Genet., 43 (2011), 95-100.

[21]

D. Huh and J. Paulsson, Random partitioning of molecules at cell division, Proc. Natl. Acad. Sci. USA, 108 (2011), 15004-15009. doi: 10.1073/pnas.1013171108.

[22]

J. JaruszewiczM. Kimmel and T. Lipniacki, Stability of bacterial toggle switches is enhanced by cell-cycle lengthening by several orders of magnitude, Phys. Rev. E., 89 (2014), 022710. doi: 10.1103/PhysRevE.89.022710.

[23]

F. JiaoM. X. Tang and J. S. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Differ. Equations, 254 (2013), 3307-3328. doi: 10.1016/j.jde.2013.01.019.

[24]

F. JiaoM. X. TangJ. S. Yu and B. Zheng, Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math., 75 (2015), 2396-2420. doi: 10.1137/151005567.

[25]

I. G. Johnston and N. S. Jones, Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions, Proc. R. Soc. A, 471 (2015), 20150050, 19pp. doi: 10.1098/rspa.2015.0050.

[26]

M. KærnT. C. ElstonW. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet., 6 (2005), 451-464.

[27]

S. J. KronC. A. Styles and G. R. Fink, Symmetric cell division in pseudohyphae of the yeast Saccharomyces cerevisiae, Mol. Biol. Cell, 5 (1994), 933-1063. doi: 10.1091/mbc.5.9.1003.

[28]

J. H. KuangM. X. Tang and J. S. Yu, The mean and noise of protein numbers in stochastic gene expression, J. Math. Biol., 67 (2013), 261-291. doi: 10.1007/s00285-012-0551-8.

[29]

E. KussellR. KishonyN. Q. Balaban and S. Leibler, Bacterial persistence: A model of survival in changing environments, Genetics, 169 (2005), 1807-1814. doi: 10.1534/genetics.104.035352.

[30]

K. Lewis, Persister cells, Annu. Rev. Microbiol., 64 (2010), 357-372. doi: 10.1146/annurev.micro.112408.134306.

[31]

Q. Y. LiL. F. Huang and J. S. Yu, Modulation of first-passage time for bursty gene expression via random signals, Math. Biosci. Eng., 14 (2017), 1261-1277. doi: 10.3934/mbe.2017065.

[32]

Y. Y. LiM. X. Tang and J. S. Yu, Transcription dynamics of inducible genes modulated by negative regulations, Math. Med. Biol., 32 (2015), 115-136. doi: 10.1093/imammb/dqt019.

[33]

E. LibbyT. J. Perkins and P. S. Swain, Noisy information processing through transcriptional regulation, Proc. Natl. Acad. Sci. USA, 104 (2007), 7151-7156. doi: 10.1073/pnas.0608963104.

[34]

J. Lloyd-PriceH. Tran and A. S. Ribeiro, Dynamics of small genetic circuits subject to stochastic partitioning in cell division, J. Theor. Biol., 356 (2014), 11-19. doi: 10.1016/j.jtbi.2014.04.018.

[35]

R. Losick and C. Desplan, Stochasticity and cell fate, Science, 320 (2008), 65-68. doi: 10.1126/science.1147888.

[36]

A. A. Martinez and J. M. Brickman, Gene expression heterogeneities in embryonic stem cell populations: Origin and function, Curr. Opin. Cell Biol., 23 (2011), 650-656. doi: 10.1016/j.ceb.2011.09.007.

[37]

B. MunskyG. Neuert and O. A. Van, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187. doi: 10.1126/science.1216379.

[38]

M. OsellaE. Nugent and L. M. Cosentino, Concerted control of Escherichia coli cell division, Proc. Natl. Acad. Sci. USA, 111 (2014), 3431-3435.

[39]

J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. doi: 10.1006/tpbi.1995.1027.

[40]

A Raj and O. A. Van, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell, 135 (2008), 216-226. doi: 10.1016/j.cell.2008.09.050.

[41]

A. SanchezS. Choubey and J. Kondev, Regulation of noise in gene expression, Annu. Rev. Biophys., 42 (2013), 469-491. doi: 10.1146/annurev-biophys-083012-130401.

[42]

A. Singh and L. S. Weinberger, Stochastic gene expression as a molecular switch for viral latency, Curr. Opin. Microbiol., 12 (2009), 460-466. doi: 10.1016/j.mib.2009.06.016.

[43]

S. O. Skinner, et al., Single-cell analysis of transcription kinetics across the cell cycle, eLife, 5 (2016), e12175. doi: 10.7554/eLife.12175.

[44]

L. H. So, et al., General properties of transcriptional time series in Escherichia coli, Nat. Genet., 43 (2011), 554–560. doi: 10.1038/ng.821.

[45]

S. L. Spencer, et al., Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis, Nature, 459 (2009), 428–432. doi: 10.1038/nature08012.

[46]

Q. W. SunM. X. Tang and J. S. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494. doi: 10.1007/s00285-011-0420-x.

[47]

Q. W. SunM. X. Tang and J. S. Yu, Temporal profile of gene transcription noise modulated by cross-talking signal transduction pathways, B. Math. Biol., 74 (2012), 375-398. doi: 10.1007/s11538-011-9683-z.

[48]

P. S. SwainM. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci. USA, 99 (2002), 12795-12800.

[49]

M. X. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol., 253 (2008), 271-280. doi: 10.1016/j.jtbi.2008.03.023.

[50]

M. L. TurnerE. D. Hawkins and P. D. Hodgkin, Quantitative regulation of B cell division destiny by signal strength, J. Immunol., 181 (2008), 374-382. doi: 10.4049/jimmunol.181.1.374.

[51]

Y. VoichekR. Bar-Ziv and N. Barkai, Expression homeostasis during DNA replication, Science, 351 (2016), 1087-1090. doi: 10.1126/science.aad1162.

[52]

H. H. WangZ. J. YuanP. J. Liu and T. S. Zhou, Division time-based amplifiers for stochastic gene expression, Mol. Biosyst., 11 (2015), 2417-2428. doi: 10.1039/C5MB00391A.

[53]

L. S. Weinberger, et al., Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity, Cell, 122 (2005), 169–182. doi: 10.1016/j.cell.2005.06.006.

[54]

J. S. Yu and X. J. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.

[55]

J. S. YuQ. W. Sun and M. X. Tang, The nonlinear dynamics and fluctuations of mRNA levels in cross-talking pathway activated transcription, J. Theor. Biol., 363 (2014), 223-234. doi: 10.1016/j.jtbi.2014.08.024.

Figure 1.  Modeling of two-state stochastic model of transcription with cell division. A. Kinetic scheme for describing two-state transcription model, where G and G$'$ denote the gene is active and inactive, respectively. B. A simplified diagram intuitively illustrates the cell cycle. The $i$th cell cycle is from ${\rm W}_{i-1}$ to ${\rm W}_{i}$, the duration of the $i$th cell cycle is ${\rm T}_i$. The cell division events are indicated by arrows.
Figure 2.  Schematic diagrams for the time evolution with cell division. The cell division events are indicated by green arrows. W$_i$ stands for the $i$th cell division point, $\tau_i$ is the value of T$_i$, $t_{i+1}$ is the elapsed time since the $i$th (the recent) cell division, then $t = \sum^{i}_{j = 1}\tau_j+t_{i+1}$.
Figure 3.  Schematic diagram for cell division modes, where the arrow points to the aim daughter cell. A. Symmetric cell division. B. Binomial inheritance. C. Random subtractive inheritance. D. Random additive inheritance.
Figure 4.  Temporal changes in the mean transcript level. The red solid lines represent analytic solutions and the blue dashed lines with circles sign represent numerical solutions. A. Temporal changes in the mean transcript level with BR. B. Temporal changes in the mean transcript level with AS.
Figure 5.  Influence of mean cell-cycle length $\tau$ on the steady-state mean transcript level and steady-state mRNA noise, where the cell cycle obeys log-normal distribution. The black stars, yellow stars, magenta stars, blue stars, red stars and green stars represent the steady-state mean transcript level (the steady-state mRNA noise) with S, BF, BR, RA, AS and RS, respectively, where the dashed lines represent the fittings. A. Influence of mean cell-cycle length $\tau$ on the steady-state mean transcript level. B. Influence of mean cell-cycle length $\tau$ on the steady-state mRNA noise.
Table 1.  The steady-state mean transcript level (the steady-state mRNA noise) with S, BF, BR, and different cell cycle distributions, where the mean of the cell cycle is $\tau = 120$.
S BF BR
Constant cell cycle 86.4910 (0.0402) 86.4910 (0.0408) 86.4924 (0.0408)
Exponential distribution 87.2826 (0.0395) 87.2826 (0.0401) 87.2840 (0.0401)
Log-normal distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
Erlang distribution 86.8273 (0.0400) 86.8273 (0.0406) 86.8276 (0.0407)
Uniform distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
S BF BR
Constant cell cycle 86.4910 (0.0402) 86.4910 (0.0408) 86.4924 (0.0408)
Exponential distribution 87.2826 (0.0395) 87.2826 (0.0401) 87.2840 (0.0401)
Log-normal distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
Erlang distribution 86.8273 (0.0400) 86.8273 (0.0406) 86.8276 (0.0407)
Uniform distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
Table 2.  The steady-state mean transcript level (the steady-state mRNA noise) with RA, AS, RS and different cell cycle distributions, where the mean of the cell cycle is $\tau = 120$.
RA AS RS
Constant cell cycle 96.0226 (0.0353) 94.1360 (0.0367) 92.2314 (0.0381)
Exponential distribution 95.9311 (0.0356) 94.1327 (0.0367) 92.3301 (0.0379)
Log-normal distribution 95.7876 (0.0354) 94.1105 (0.0367) 92.4612 (0.0379)
Erlang distribution 95.9445 (0.0354) 94.1248 (0.0367) 92.3034 (0.0380)
Uniform distribution 95.9698 (0.0353) 94.1229 (0.0367) 92.2841 (0.0381)
RA AS RS
Constant cell cycle 96.0226 (0.0353) 94.1360 (0.0367) 92.2314 (0.0381)
Exponential distribution 95.9311 (0.0356) 94.1327 (0.0367) 92.3301 (0.0379)
Log-normal distribution 95.7876 (0.0354) 94.1105 (0.0367) 92.4612 (0.0379)
Erlang distribution 95.9445 (0.0354) 94.1248 (0.0367) 92.3034 (0.0380)
Uniform distribution 95.9698 (0.0353) 94.1229 (0.0367) 92.2841 (0.0381)
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